Gamers want to compare the probabilities of the possible dice throws used in
their role-playing games. Invoking the combinatorics of balls and urns, Doctor Vogler
confirms the complexity of enumerating the possibilities for non-trivial numbers of
rolls and dice.

How many ways can six people share four rooms that each sleeps up to four? After
carefully interpreting the question, Doctors Ian and Peterson help a combinatorics
student think through the possibilities.

At least how many balance scale weighings of ten coins do you need to determine the
two fakes? By applying combinatorics and keeping track of lower bounds, Doctor
Jacques provides a methodical approach.

How many ways are there to make dice out of the Platonic solids (i.e. 4,
6, 8, 12, and 20 sides)? How many of those ways have opposite face sums
equal? What would the opposing face sums be for each type?

Nine consecutive spins of a roulette wheel surprise a gambler unfamiliar with how to
determine the likelihood of independent events. Doctor Vogler obliges with the
requested combinations and probabilities, but only after questioning our arbitrary
preference for some events over others that may have the same -- or even longer --
odds.